Praxis 5018 Math
The Mathematics content category has about 41 questions which
make up about 29% of the test.
There are four Mathematics sections:
- Number and Operations
- Algebraic Thinking
- Geometry and Measurement
- Data, Statistics, and Probability
So, let’s start with Number and Operations.
Number and Operations
This section tests your knowledge on the base-ten number system, place value, basic mathematical processes, and the order of operations.
Let’s look at some concepts that may pop up on the test.
Expanded Form
Expanded form is when a number is expressed by breaking the number down into the value of each digit. For example, the number 492 means that there are 4 hundreds, 9 tens, and 2 ones, so the expanded form of 492 would be:
400 + 90 + 2 = 492
Expanded form is an important part of understanding place value because students need to understand what each digit in a number is actually worth.
Other examples of expanded form are:
1,000 + 900 +30 + 4 = 1,934
70 + 3 = 73
800 + 50 + 2 = 852
Place Value
Place value is the value of a digit based on where that digit is in the number. A digit is just a single whole number (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) in a number. For example, the number 29 has two digits (2 and 9). The number 138 has 3 digits (1, 3, and 8).
The place where a digit is located in a number determines its value. A place value chart, such as the one below, can help students understand how many ones, tens, hundreds, etc. are in a number.
For example, the number 4,712 has a 4 in the thousands place, a 7 in the hundreds place, a 1 in the tens place, and a 2 in the ones place. This means there are 4 thousands, 7 hundreds, 1 ten, and 4 ones in this number. The value of the 4 in the thousands place is 4,000 because the digit 4 is actually worth 4,000, not just 4 ones. Models and place value blocks can be helpful when teaching students about place value.
Here is an example of a place value chart for the number 2,356,703:
Order of Operations
The order of operations is the order in which computations must be completed in a math problem. The order of operations is as follows:
- Parenthesis
- Exponents
- Multiplication & Division
- Addition & Subtraction
This means that in a given math expression, anything that is contained within parenthesis must be completed first. After solving the expressions in the parenthesis, you would solve any part of the expression with an exponent. After the exponents, you would solve any multiplication or division portion of the problem, in the order they appear in the problem moving left to right. The last step is to solve any addition or subtraction parts of the problem, again moving left to right.
A common misconception about the order of operations is that multiplication comes before division and addition comes before subtraction. This is not the case. When you are at the multiplication and division step, you will solve whichever one comes first when you read the problem from left to right. The same thing applies to the addition and subtraction step. The order of operations can be remembered by the acronym PEMDAS.
Let’s work an example together:
30 – (7 – 3) x 5 + 4
- (7 – 3) would be solved first, because it is contained within parenthesis. The expression would now be: 30 – 4 x 5 + 4
- There are no exponents in this expression, so you would move on to multiplication and division and solve 4 x 5. The expression would now be: 30 – 20 + 4.
- The next step is addition and subtraction. Since the subtraction occurs first in this problem when read from left to right, you would do that first. The expression would now be 10 + 4.
- To complete the problem, you would solve 10 + 4 to get an answer of 14.
Properties of Operations
The properties of operations include the commutative property, associative property, and the distributive property. These properties refer to various strategies used to solve math equations.
The commutative property means that numbers can be added or multiplied in any order, and the answer will still be the same. For example, 4 + 7 + 3 will get the same result as 7 + 3 + 4. Similarly, 5 x 2 x 3 will get the same answer as 3 x 2 x 5. The commutative property does not apply to subtraction or division, because the answer will not be the same when the numbers are in a different order. For example, 5 – 2 will not get the same answer as 2 – 5.
The associative property means that numbers can be added or multiplied even when grouped in different ways, and the answer will still be the same. “Grouped” means put into parentheses for the purpose of the problem. For example, (6 x 2) x 4 will get the same answer as 6 x (2 x 4). Both expressions have a solution of 48. As with the commutative property, this does not apply to subtraction or division.
The distributive property means that you can multiply a number by the numbers or variables within the parentheses, and then find the sum. An example of the distributive property is shown below:
3(4 + 2)
Rather than doing 4 + 2 then multiplying this by 3, we can also do 3 x 4 then
3 x 2, and then add the two results:
3 x 4 + 3 x 2
12 + 6
18
The distributive property is most often used in problems with variables. An example of this is:
6(y + 4)
6y + 6 x 4
6y + 24
This section tests your knowledge on the base-ten number system, place value, basic mathematical processes, and the order of operations.
Let’s look at some concepts that may pop up on the test.
Expanded Form
Expanded form is when a number is expressed by breaking the number down into the value of each digit. For example, the number 492 means that there are 4 hundreds, 9 tens, and 2 ones, so the expanded form of 492 would be:
400 + 90 + 2 = 492
Expanded form is an important part of understanding place value because students need to understand what each digit in a number is actually worth.
Other examples of expanded form are:
1,000 + 900 +30 + 4 = 1,934
70 + 3 = 73
800 + 50 + 2 = 852
Place Value
Place value is the value of a digit based on where that digit is in the number. A digit is just a single whole number (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) in a number. For example, the number 29 has two digits (2 and 9). The number 138 has 3 digits (1, 3, and 8).
The place where a digit is located in a number determines its value. A place value chart, such as the one below, can help students understand how many ones, tens, hundreds, etc. are in a number.
For example, the number 4,712 has a 4 in the thousands place, a 7 in the hundreds place, a 1 in the tens place, and a 2 in the ones place. This means there are 4 thousands, 7 hundreds, 1 ten, and 4 ones in this number. The value of the 4 in the thousands place is 4,000 because the digit 4 is actually worth 4,000, not just 4 ones. Models and place value blocks can be helpful when teaching students about place value.
Here is an example of a place value chart for the number 2,356,703:
Order of Operations
The order of operations is the order in which computations must be completed in a math problem. The order of operations is as follows:
- Parenthesis
- Exponents
- Multiplication & Division
- Addition & Subtraction
This means that in a given math expression, anything that is contained within parenthesis must be completed first. After solving the expressions in the parenthesis, you would solve any part of the expression with an exponent. After the exponents, you would solve any multiplication or division portion of the problem, in the order they appear in the problem moving left to right. The last step is to solve any addition or subtraction parts of the problem, again moving left to right.
A common misconception about the order of operations is that multiplication comes before division and addition comes before subtraction. This is not the case. When you are at the multiplication and division step, you will solve whichever one comes first when you read the problem from left to right. The same thing applies to the addition and subtraction step. The order of operations can be remembered by the acronym PEMDAS.
Let’s work an example together:
30 – (7 – 3) x 5 + 4
- (7 – 3) would be solved first, because it is contained within parenthesis. The expression would now be: 30 – 4 x 5 + 4
- There are no exponents in this expression, so you would move on to multiplication and division and solve 4 x 5. The expression would now be: 30 – 20 + 4.
- The next step is addition and subtraction. Since the subtraction occurs first in this problem when read from left to right, you would do that first. The expression would now be 10 + 4.
- To complete the problem, you would solve 10 + 4 to get an answer of 14.
Properties of Operations
The properties of operations include the commutative property, associative property, and the distributive property. These properties refer to various strategies used to solve math equations.
The commutative property means that numbers can be added or multiplied in any order, and the answer will still be the same. For example, 4 + 7 + 3 will get the same result as 7 + 3 + 4. Similarly, 5 x 2 x 3 will get the same answer as 3 x 2 x 5. The commutative property does not apply to subtraction or division, because the answer will not be the same when the numbers are in a different order. For example, 5 – 2 will not get the same answer as 2 – 5.
The associative property means that numbers can be added or multiplied even when grouped in different ways, and the answer will still be the same. “Grouped” means put into parentheses for the purpose of the problem. For example, (6 x 2) x 4 will get the same answer as 6 x (2 x 4). Both expressions have a solution of 48. As with the commutative property, this does not apply to subtraction or division.
The distributive property means that you can multiply a number by the numbers or variables within the parentheses, and then find the sum. An example of the distributive property is shown below:
3(4 + 2)
Rather than doing 4 + 2 then multiplying this by 3, we can also do 3 x 4 then
3 x 2, and then add the two results:
3 x 4 + 3 x 2
12 + 6
18
The distributive property is most often used in problems with variables. An example of this is:
6(y + 4)
6y + 6 x 4
6y + 24
Algebraic Thinking
This section tests your knowledge on solving algebraic equations and your ability to analyze and interpret these solutions.
Let’s take a look at some concepts that you may see on the test.
Expression versus Equation
The main difference between an expression and equation is whether or not there is an equal sign. An equation contains an equal sign, while an expression does not. Both equations and expressions can have a solution, or answer, but only an equation will have an equal sign. For example:
16 + 4 = 20 is an equation.
16 + 4 is an expression.
Solving for x
Solving for x in an equation involves getting the x by itself on one side of the equation, leaving the answer (or value of x) on the other side. To do this, you will eliminate anything on the side with the x, one step at a time, in the
reverse
order of the order of operations. Since an equation must stay balanced, or equal, anything that you do to one side must also be done to the other side. Let’s look at an example, with explanations for each step:
3x + 4 = 13
To get x by itself on one side, we must get rid of the 4 that is being added, and the 3 that is being multiplied by x. Since the order of operations
ends
with addition and subtraction, and we are doing the reverse order, we will start by getting rid of the + 4 by doing the opposite of addition (subtraction). To do this, you will subtract 4 from each side of the equation:
3x + 4 – 4
= 13 – 4
This gets us to:
3x = 9
The 3 is still on the side of the equation with the x. Since the 3 is being multiplied by the x, we will do the opposite of multiplication and divide each side by 3:
3x ÷ 3
= 9 ÷ 3
3 divided by 3 is 1, but you do not need to put a 1 in front of the x since 1 times x would still be x. This now leaves us with the x on one side of the equation and 3 on the other side, meaning x is equal to 3:
x = 3
You can then check your work by putting this value for x back into the original equation to see if both sides are equal:
3(3) + 4 = 13
9 + 4 = 13
13 = 13
Since both sides of the equation are 13, we know our answer is correct.
Dependent versus Independent Variables
Independent and dependent variables are the variables used in data or an experiment. There are several ways to think about dependent and independent variables, but the main difference is that the dependent variable
depends on
the independent variable. The independent variable is the one that
you
can change, and the dependent variable is the one that will change based on the value of the independent variable.
Another way to think of this is to think of the independent variable as being the cause and the dependent variable as being the effect. In a set of data, the x-value will be the independent variable, and the y-value will be the dependent variable. For example, if a toy costs $5 and you are asked to find the total cost based on how many toys you buy, the equation used for this would be:
y = 5x
In this equation,
y
is the total cost of the toys, and
x
is the number of toys you buy. Since you control the number of toys you buy, this is an independent variable. The y-value (or total cost), however,
depends on
the number of toys you buy.
This section tests your knowledge on solving algebraic equations and your ability to analyze and interpret these solutions.
Let’s take a look at some concepts that you may see on the test.
Expression versus Equation
The main difference between an expression and equation is whether or not there is an equal sign. An equation contains an equal sign, while an expression does not. Both equations and expressions can have a solution, or answer, but only an equation will have an equal sign. For example:
16 + 4 = 20 is an equation.
16 + 4 is an expression.
Solving for x
Solving for x in an equation involves getting the x by itself on one side of the equation, leaving the answer (or value of x) on the other side. To do this, you will eliminate anything on the side with the x, one step at a time, in the
reverse
order of the order of operations. Since an equation must stay balanced, or equal, anything that you do to one side must also be done to the other side. Let’s look at an example, with explanations for each step:
3x + 4 = 13
To get x by itself on one side, we must get rid of the 4 that is being added, and the 3 that is being multiplied by x. Since the order of operations
ends
with addition and subtraction, and we are doing the reverse order, we will start by getting rid of the + 4 by doing the opposite of addition (subtraction). To do this, you will subtract 4 from each side of the equation:
3x + 4 – 4
= 13 – 4
This gets us to:
3x = 9
The 3 is still on the side of the equation with the x. Since the 3 is being multiplied by the x, we will do the opposite of multiplication and divide each side by 3:
3x ÷ 3
= 9 ÷ 3
3 divided by 3 is 1, but you do not need to put a 1 in front of the x since 1 times x would still be x. This now leaves us with the x on one side of the equation and 3 on the other side, meaning x is equal to 3:
x = 3
You can then check your work by putting this value for x back into the original equation to see if both sides are equal:
3(3) + 4 = 13
9 + 4 = 13
13 = 13
Since both sides of the equation are 13, we know our answer is correct.
Dependent versus Independent Variables
Independent and dependent variables are the variables used in data or an experiment. There are several ways to think about dependent and independent variables, but the main difference is that the dependent variable
depends on
the independent variable. The independent variable is the one that
you
can change, and the dependent variable is the one that will change based on the value of the independent variable.
Another way to think of this is to think of the independent variable as being the cause and the dependent variable as being the effect. In a set of data, the x-value will be the independent variable, and the y-value will be the dependent variable. For example, if a toy costs $5 and you are asked to find the total cost based on how many toys you buy, the equation used for this would be:
y = 5x
In this equation,
y
is the total cost of the toys, and
x
is the number of toys you buy. Since you control the number of toys you buy, this is an independent variable. The y-value (or total cost), however,
depends on
the number of toys you buy.
Geometry and Measurement
This section tests your knowledge on measurement and analysis of one-, two-, and three-dimensional figures and your knowledge on graphing and analyzing data using a coordinate plane.
Take a look at these concepts.
Surface Area
Surface area is the total area of all surfaces on a three-dimensional figure. Rather than memorizing formulas for surface areas of three-dimensional figures, it is easier to think about finding the area of each side, then adding these areas together for the total surface area.
Some basic formulas for areas of two-dimensional figures that might be helpful are:
Area of a square or rectangle = length x width
Area of a triangle = ½(base x height)
Area of a circle =
r
²
To find the surface of a rectangular prism, you find the area of each of the 6 sides and then add these values together. An example of this is shown below:
To find the surface area of this rectangular prism, you need to first find the area of each side. The area of the top and bottom are each 16
cm
2
, since 4 x 4 = 16. The area of the four remaining sides are each 24
cm
², since 6 x 4 = 24. To find the total surface area, we need to add the area of each side together:
16 + 16 + 24 + 24 + 24 + 24 = 128
So, this rectangular prism has a total surface area of 128
cm²
.
Coordinate Plane
A coordinate plane is used as a way to show a visual representation of data points or an equation with variables. A coordinate plane is formed by an x-axis that runs horizontally (side to side) and a y-axis that runs vertically (up and down). The point where these two axes intersect is called the
origin
and has a coordinate of (0,0).
A coordinate is shown as two numbers in parenthesis, divided by a comma. The first number is the x-value and the second number is the y-value. The x-value shows how far over to go on the x-axis. The y-value shows how high up to go on the y-axis. The data point on the graph will be where these two lines meet. The coordinate plane below shows three examples of different data points.
A coordinate plane is divided into four quadrants. The four quadrants are labeled below:
Mass
Mass is how much matter is contained in an object. Although the actual definition of mass is slightly different than weight, mass and weight are very similar and mass is what we typically think of as weight. Mass is found by multiplying the volume of an object by its density. The following formula is used for mass:
Mass = Density x Volume
If an object has a volume of 50
cm
³
and a density of 2 grams/
cm³
, the mass would be 100 grams (50 x 2).
For general purposes, since mass typically has the same value as weight, you can also use a scale or balance to find an object’s mass.
Rays
A ray can be thought of as a straight line that has a starting point, but no ending point. A ray starts at a specific point but then continues in a certain direction to infinity. Rays are shown by a dot at the starting point and a line ending in an arrow, indicating that the line continues on with no ending point. Two rays can have the same starting point and then go in different directions. The coordinate plane below shows an example of two rays with the same starting point:
This section tests your knowledge on measurement and analysis of one-, two-, and three-dimensional figures and your knowledge on graphing and analyzing data using a coordinate plane.
Take a look at these concepts.
Surface Area
Surface area is the total area of all surfaces on a three-dimensional figure. Rather than memorizing formulas for surface areas of three-dimensional figures, it is easier to think about finding the area of each side, then adding these areas together for the total surface area.
Some basic formulas for areas of two-dimensional figures that might be helpful are:
Area of a square or rectangle = length x width
Area of a triangle = ½(base x height)
Area of a circle =r²
To find the surface of a rectangular prism, you find the area of each of the 6 sides and then add these values together. An example of this is shown below:
To find the surface area of this rectangular prism, you need to first find the area of each side. The area of the top and bottom are each 16
cm
2
, since 4 x 4 = 16. The area of the four remaining sides are each 24
cm
², since 6 x 4 = 24. To find the total surface area, we need to add the area of each side together:
16 + 16 + 24 + 24 + 24 + 24 = 128
So, this rectangular prism has a total surface area of 128 cm².
Coordinate Plane
A coordinate plane is used as a way to show a visual representation of data points or an equation with variables. A coordinate plane is formed by an x-axis that runs horizontally (side to side) and a y-axis that runs vertically (up and down). The point where these two axes intersect is called the
origin
and has a coordinate of (0,0).
A coordinate is shown as two numbers in parenthesis, divided by a comma. The first number is the x-value and the second number is the y-value. The x-value shows how far over to go on the x-axis. The y-value shows how high up to go on the y-axis. The data point on the graph will be where these two lines meet. The coordinate plane below shows three examples of different data points.
A coordinate plane is divided into four quadrants. The four quadrants are labeled below:
Mass
Mass is how much matter is contained in an object. Although the actual definition of mass is slightly different than weight, mass and weight are very similar and mass is what we typically think of as weight. Mass is found by multiplying the volume of an object by its density. The following formula is used for mass:
Mass = Density x Volume
If an object has a volume of 50cm³ and a density of 2 grams/cm³, the mass would be 100 grams (50 x 2).
For general purposes, since mass typically has the same value as weight, you can also use a scale or balance to find an object’s mass.
Rays
A ray can be thought of as a straight line that has a starting point, but no ending point. A ray starts at a specific point but then continues in a certain direction to infinity. Rays are shown by a dot at the starting point and a line ending in an arrow, indicating that the line continues on with no ending point. Two rays can have the same starting point and then go in different directions. The coordinate plane below shows an example of two rays with the same starting point:
Data, Statistics, and Probability
This section tests your knowledge on basic statistics, data, and methods used to interpret this data.
Here are some concepts that are likely to pop up on the test.
Measures of Central Tendency
A measure of central tendency is a way to identify a typical value for a set of data. Ways to measure central tendency include mean, median, and mode:
Mean
is another word for average. In order to find the mean for a set of numbers, you need to find the sum (or total) of all of the numbers, then divide that sum by the amount of numbers or values in the data set.
For example, to find the mean of 98, 95, and 83 you would add 98 + 95 + 83 to get 276. You would then divide 276 by 3 because there are 3 different numbers (98, 95, and 93). 276 divided by 3 equals 92, so the mean of this set of data is 92.
Median
is the middle value when a set of numbers are put in order from least to greatest. If there is an even amount of numbers and two numbers are in the middle, you would find the average of those two numbers.
For example, to find the median of 34, 33, 38, 37, and 29, you would need to arrange the numbers in order from least to greatest:
29, 33, 34, 37, 38
Since 34 is in the middle, 34 is the median.
The following data set has two numbers that are in the middle:
12, 14, 15, 17, 20, 21
So, you would find the average of 15 and 17 to get a median of 16.
Mode
is the number that appears most frequently in a set of numbers. For example, the mode of the following set of numbers is 18, because it appears 4 times in the set while other numbers appear one, two, or three times:
13, 10, 13, 18, 12, 12, 18, 18, 12, 18
If no number is repeated in a set, then that set of data has no mode.
A set of data can also have more than one mode if more than one number appears most frequently.
Probability
Probability means the likelihood of something happening. In order to determine the probability of something happening, you need to look at the total possible outcomes and the number of chances you have for one specific outcome occurring.
For example, if there are 3 pink pencils, 2 yellow pencils, and 5 blue pencils in a bag, there are 10 potential outcomes (10 different pencils). If you want to know the probability of choosing a pink pencil, you would set up a ratio, or fraction, showing the chance of choosing a pink pencil. Since there are 3 pink pencils and 10 pencils total, the probability of choosing a pink pencil would be 3/10 (or 3 out of 10).
And that’s some basic info about the Mathematics content category.
This section tests your knowledge on basic statistics, data, and methods used to interpret this data.
Here are some concepts that are likely to pop up on the test.
Measures of Central Tendency
A measure of central tendency is a way to identify a typical value for a set of data. Ways to measure central tendency include mean, median, and mode:
Mean
is another word for average. In order to find the mean for a set of numbers, you need to find the sum (or total) of all of the numbers, then divide that sum by the amount of numbers or values in the data set.
For example, to find the mean of 98, 95, and 83 you would add 98 + 95 + 83 to get 276. You would then divide 276 by 3 because there are 3 different numbers (98, 95, and 93). 276 divided by 3 equals 92, so the mean of this set of data is 92.
Median
is the middle value when a set of numbers are put in order from least to greatest. If there is an even amount of numbers and two numbers are in the middle, you would find the average of those two numbers.
For example, to find the median of 34, 33, 38, 37, and 29, you would need to arrange the numbers in order from least to greatest:
29, 33, 34, 37, 38
Since 34 is in the middle, 34 is the median.
The following data set has two numbers that are in the middle:
12, 14, 15, 17, 20, 21
So, you would find the average of 15 and 17 to get a median of 16.
Mode
is the number that appears most frequently in a set of numbers. For example, the mode of the following set of numbers is 18, because it appears 4 times in the set while other numbers appear one, two, or three times:
13, 10, 13, 18, 12, 12, 18, 18, 12, 18
If no number is repeated in a set, then that set of data has no mode.
A set of data can also have more than one mode if more than one number appears most frequently.
Probability
Probability means the likelihood of something happening. In order to determine the probability of something happening, you need to look at the total possible outcomes and the number of chances you have for one specific outcome occurring.
For example, if there are 3 pink pencils, 2 yellow pencils, and 5 blue pencils in a bag, there are 10 potential outcomes (10 different pencils). If you want to know the probability of choosing a pink pencil, you would set up a ratio, or fraction, showing the chance of choosing a pink pencil. Since there are 3 pink pencils and 10 pencils total, the probability of choosing a pink pencil would be 3/10 (or 3 out of 10).
And that’s some basic info about the Praxis 5018 Mathematics content category.